Title:
Heavy-quark spin symmetry partners of the $X(3872)$ revisited

Abstract: We revisit the consequences of the heavy-quark spin symmetry for the possible
spin partners of the $X(3872)$. We confirm that, if the $X(3872)$ were a
$D\bar{D}^*$ molecular state with the quantum numbers $J^{PC}=1^{++}$, then in
the strict heavy-quark limit there should exist three more hadronic molecules
degenerate with the $X(3872)$, with the quantum numbers $0^{++}$, $1^{+-}$, and
$2^{++}$ in line with previous results reported in the literature. We
demonstrate that this result is robust with respect to the inclusion of the
one-pion exchange interaction between the $D$ mesons. However, this is true
only if all relevant partial waves as well as particle channels which are
coupled via the pion-exchange potential are taken into account. Otherwise, the
heavy-quark symmetry is destroyed even in the heavy-quark limit. Finally, we
solve the coupled-channel problem in the $2^{++}$ channel with nonperturbative
pions beyond the heavy-quark limit and, contrary to the findings of previous
calculations with perturbative pions, find for the spin-2 partner of the
$X(3872)$ a significant shift of the mass as well as a width of the order of 50
MeV.